Oscillation Calculator - Harmonic Motion

This physics calculator helps you calculate oscillation parameters for spring-mass systems including period, frequency, and amplitude. Simple harmonic motion is a fundamental concept in physics.

Oscillation Definition

• Simple Harmonic Motion: Periodic motion with restoring force proportional to displacement
• Natural Frequency: ω₀ = √(k/m) (undamped frequency)
• Period: T = 2π/ω₀ (time for one complete cycle)
• Frequency: f = ω₀/(2π) (cycles per second)

Oscillation Formulas

• Natural Frequency: ω₀ = √(k/m)
• Period: T = 2π/ω₀ = 2π√(m/k)
• Frequency: f = ω₀/(2π) = (1/2π)√(k/m)
• Damping Ratio: ζ = c/c_c = c/(2√(mk))
• Damped Frequency: ω_d = ω₀√(1-ζ²)
• Maximum Velocity: v_max = Aω₀
• Maximum Acceleration: a_max = Aω₀²

Applications

• Mechanics: Springs, pendulums, and vibrating systems
• Electronics: LC circuits and oscillators
• Engineering: Vibration analysis and control
• Acoustics: Sound waves and musical instruments
• Optics: Light waves and lasers
• Biology: Heartbeat and circadian rhythms

Important Notes

• Period is independent of amplitude (for simple harmonic motion)
• Energy is conserved in undamped systems
• Damping reduces amplitude over time
• Critical damping prevents oscillation
• Resonance occurs at natural frequency

Oscillation Types

• Undamped: No energy loss, constant amplitude
• Underdamped: Oscillating with decreasing amplitude
• Critically Damped: Fastest return to equilibrium
• Overdamped: Slow return without oscillation
• Forced: Driven by external force

Oscillation Examples

• Spring-Mass: Mass on spring
• Pendulum: Swinging mass
• LC Circuit: Capacitor and inductor
• Molecular: Atomic vibrations
• Structural: Building vibrations

Energy in Oscillation

• Kinetic Energy: Maximum at equilibrium
• Potential Energy: Maximum at extremes
• Total Energy: Constant in undamped system
• Energy Loss: Due to damping
• Power Input: Required for forced oscillation

Damping Effects

• Light Damping: ζ < 1, oscillating decay
• Critical Damping: ζ = 1, fastest settling
• Heavy Damping: ζ > 1, no oscillation
• Logarithmic Decrement: δ = ln(A₁/A₂)
• Quality Factor: Q = 1/(2ζ)

Resonance

• Natural Frequency: System's preferred frequency
• Resonance: Maximum response at natural frequency
• Bandwidth: Frequency range of significant response
• Q Factor: Sharpness of resonance peak
• Phase Shift: 90° at natural frequency